Freedom of Speech Essay - 2651 Words

Freedom of Speech Freedom of speech is something that, traditionally, has been unique to the United States. In fact, it was one of the founding principles of our country. It was the group of repressed colonists that decided that they would no longer accept the fact that they were not allowed to speak out against the throne. They decided that they would not accept not being able to denounce the Church of England. And on July 4, 1776, what could be seen as one of the boldest forms of speech ever ensured that never again would Americans be subject to the harsh regulation of Great Britain. Then, in 1789, our nations forefathers made it official with the ratification of the U.S. Constitution, including the first of ten amendments:†¦show more content†¦Is it ethical for the United States to even request that the rest of the world join our standard? Finally, is it ethical for the United States to do its own thing, regardless of others? Each of these concerns will be examined and the answer to all of them is no, it is not ethical - but what other choice is there? In the United States, as mentioned above, Americans enjoy a unique freedom - the freedom to express themselves as they wish. But freedom does not come without limits. The first amendment does not protect speech that is obscene and/or profane. But exactly what is obscene? What is profane? It honestly depends on who you ask. Even when brought before the United States Supreme Court, Justices could not define it. Justice Potter Steward couldnt do it either, but said I know it when I see it. As Winchel Todd Vincent III reflects in his paper Freedom of Speech in Cyberspace: Are you a criminal?, obscenity and profanity are determined by community standards. For the most part, people can agree what is good and what is bad. Vincent writes, The test for obscenity is whether to the average person, applying contemporary community standards, the dominant theme of the material taken as a whole appeals to prurient interests. Prurient is defined by Websters as marked by or arousing an immoderate or unwholesome interest or desire, especially a sexual desire.2 In other countries, standards are not as free asShow MoreRelatedSpeech : Freedom Of Speech1364 Words   |  6 Pages Even though freedom of speech can be tricky, understanding what can be said under the first amendment is key. This will give you more confidence when you re talking about current events, it ll inhibit social interaction, but the biggest reason of all is understanding your rights will keep you out of jail. Even the supreme court struggles to determine what exactly constitutes protected speech. When trying to see if your covered by the first amendment, certain questions have to be displayed. AreRead MoreSpeech On Freedom Of Speech1484 Words   |  6 Pages Abby Koopmann Ethics Freedom of Speech Professor Hunt Culver Stockton College Freedom of Speech Americans have many freedoms that people in other countries can only wish they had. Just imagine a life where you could possibly be killed for speaking your mind and stating your opinion, other countries are living lives like that. According to Katy Davis, The United States ratified the first amendment on December 15th 1791 (Davis, n.d). We as a country don’t know where the governmentRead MoreFreedom Of Speech : Speech1105 Words   |  5 Pages Freedom of Speech Freedom of Speech, part of the First Amendment, is a privileged right that should not be taken lightly. The Milo Bill is said to protect students’ right to their freedom of speech on school grounds. It was introduced at Tennessee’s State House and is named after Milo Yiannopoulos, a British public speaker who made a career out of â€Å"trolling† liberals and gained publicity for uncalled-for acts, such as racist and harassing comments on Twitter, which got him banned from the socialRead MoreSpeech : The Freedom Of Speech Essay1624 Words   |  7 PagesCitizens of the United States are privileged to the freedom of speech under the First Amendment, but the constitutional limits of the freedom of speech have been questioned on multiple occasions. Citizens of the United States have called upon the Supreme Court numerous times to interpret the meaning of the First Amendment, and the court has censored some forms of speech such as obscene speech --which has been pr ohibited--and indecent or pornographic speech--which has been regulated (Barrett, 1999). PublicRead MoreSpeech On Freedom Of Speech3937 Words   |  16 PagesZhiyuan Li Philosophy 2367 Freedom of Speech A couple of months ago, University of Chicago’s president Robert J. Zimmerin expressed his concern about freedom of speech on campus in a Wall Street Journal article: Free speech is at risk at the very institution where it should be assured: the university †¦ Invited speakers are disinvited because a segment of a university community deems them offensive, while other orators are shouted down for similar reasons. Demands are made to eliminate readings thatRead MoreFree Speech : Freedom Of Speech1403 Words   |  6 PagesFREE SPEECH Freedom of speech is perhaps the most precious tenets of liberal democracies accorded to the citizens of the free world. The United States, and the rest of the Western society is known to be the cradle of democratic principles, and this can be evidenced by the level of freedom accorded to the citizens of the United States and the West in terms of speech. The freedom of speech gives an individual an opportunity to speak his or her mind and also give his or her opinion, sometimes on veryRead MoreFreedom Of The Media And Freedom Of Speech1540 Words   |  7 Pageswhen Socrates was forced to drink poison for his â€Å"corruption of youth† (Mette), which can be seen as silencing one man for the betterment of everyone, but there is always an ample amount of opinions on such a controversial topic as Freedom of the press, or Freedom of Speech. Many nations today believe that very strict and regulated system of governing the media and entertainment is the best answer, such as China, w here â€Å"censorship was considered a legitimate instrument for regulating the moral and politicalRead MoreFree Speech Or Freedom Of Speech Essay911 Words   |  4 PagesIntroduction America is known as The Land of Freedom, basically we are entitled for legal freedom to do whatever we wanted to. However, the freedom of speech has been challenged by many university students and educators recently. In their argument, they believe that free speech has triggered students’ unhappiness and jeopardize campus’ safety. What is the free speech or freedom of speech? According to dictionary, freedom of speech is the right of people to express their opinions publicly withoutRead MoreFreedom of Speech3842 Words   |  16 PagesFreedom of speech Freedom of speech  is the freedom to speak freely without  censorship. The synonymous term  freedom of expression  is sometimes used to indicate not only freedom of verbal speech but any act of seeking, receiving and imparting information or ideas, regardless of the medium used. In practice, the right to freedom of speech is not absolute in any country and the right is commonly subject to limitations, such as on hate speech. The right to freedom of speech is recognized as a  humanRead MoreFreedom Of Speech : Speech And Expression898 Words   |  4 PagesWhen the words, â€Å"freedom of speech† are uttered today, it brings about much conflict and tension. The vagueness of the First Amendment is a pretty good indication of how different times were back when the Constitution and everything along with it were created. Back then, all the Founding Fathers wanted was the establish a democracy using the lessons they had learned from the tyranny of the British King. They wanted more freedom and power to the people. In the modern era, everything has changed, and

The Human Tissue Act 2008 Essay - 2256 Words

Background Currently in New Zealand the major law regarding the donation of organs is the Human Tissue Act 2008 introduced by the then labour government’s Minister of Health David Cunliffe. The bill received support from both Labour and National members and in its second reading received 115 votes for and only 3 votes against (while 3 members were not present for the vote) the three votes against can be attributed to opposition from 3 of the 4 Maori party members due to cultural values tikanga Ä -iwi, Mana and Tapu The Human Tissue Act 2008 prohibits the sale and advertisement of any human tissue which includes organs without written permission at the discretion from whomever is acting as the Minister of Health. So far the only reported case (of which all exemptions must be reported in the Gazette) is the paired kidney donation scheme allowing viable living donors that wish to donate to a family or friend that is incompatible with their kidney to donate to another who is in the same po sition. This effectively facilitates a non monetary trade between two donors to ensure the chosen recipients receive a kidney that will work for them. Currently a shortage is occurring between the supply of viable organs collected from living donors and from the deceased and the demand required from those on the waiting list. Organs that can be provided from living donors include Kidneys, liver segments, some of your lung and Pancreas but of all the organs that greatest shortage that exists isShow MoreRelatedOrgan Trafficking Essay1099 Words   |  5 Pagesa call for stricter regulation of post-mortem human tissue. It would seem obvious that what is of paramount value in life would necessarily depreciate in death. 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Corporate FinanceShine plc Company

Question: Discuss about theCorporate Financefor Shine plc Company. Answer: Introduction The given assignment discusses about some of corporate finance strategies for a company that is Shine plc. The company is considering purchasing a project of new security light. For the effective decision making, strategies of corporate finance have been laid down. The assignment discusses about the cash flow analysis that is earning after taxation but before depreciation of the new security light ahs to be calculated. After this the assignment considers an analysis over net present value and internal rate of return. At last the company considers the forward contract decision making. Cash Flows of the New Security Light Earnings after taxation, but before non cash expenses are called as the cash flow from the project. In the gives case study, by deduction all the fixed and variable expenses from the regular income of the project that is annual sales, earnings after tax has been calculated. To calculate earnings after taxation but before depreciation that is cash flow, non cash expenses that is depreciation is to be deducted (Moyer, McGuigan, Rao, 2014). In calculation of cash flow of the project, depreciation is excluded. It is done because depreciation is considered as non cash expenses for the company. Depreciation does not lead to any cash flow in the financial statements of the company. Shine plc Particulars Year 1 Year 2 Year 3 Year 4 Selling units 14000 16000 16000 14000 selling price 110 120 120 100 Sales 1540000 1920000 1920000 1400000 material 420000 480000 512000 476000 Labor 210000 240000 240000 238000 Variable overhead 70000 80000 96000 98000 Interest 120000 120000 120000 120000 Depreciation 250000 250000 250000 250000 administration overhead 100000 100000 100000 100000 fixed cost 80000 80000 80000 80000 fixed overhead 500000 500000 500000 500000 consultancy 250000 0 0 0 working capital 150000 0 0 0 all cash and non cash expenses 2150000 1850000 1898000 1862000 earnings before tax -610000 70000 22000 -462000 tax 8750 2750 earnings after tax -610000 61250 19250 -462000 The cash inflow of the project has been described in the below table as earnings after tax but before depreciation. In the given case study the initial cash outlay or initial investment is -15, 00,000. Hence the net cash inflow of new security light can be calculated as -2, 60,000. year earnings after tax depreciation earnings after tax before depreciation initial outlay -1500000 -1500000 1 -610000 250000 -360000 2 61250 250000 311250 3 19250 250000 269250 4 -462000 250000 -212000 cash flow -1508500 As it has been clear from the given table that the cash flow of the new security light comes out to be negative, hence it is suggested for Shine Plc to not to accept the new security light project (Brigham, Ehrhardt, 2007). NPV (Net Present Value) In the given case study it has been given that the company follows a capital rate of 13%. The Net present value is calculated by firstly calculating present value of cash inflows (Shapiro, 2008). After present value of cash inflows, the initial investment has to be subtracted from it. If the net present value comes out to be positive, the project should be accepted. But if, the Net present value comes out to be negative, the project should be rejected (Lasher, 2016). year earnings after tax depreciation earnings after tax before depreciation PV factor @ 13% Present Value initial outlay -1500000 -1500000 -1500000 1 -610000 250000 -360000 0.885 -318600 2 61250 250000 311250 0.7831 243739.875 3 19250 250000 269250 0.6931 186617.175 4 -462000 250000 -212000 0.6133 -130019.6 cash flow -1508500 total present value -18262.55 Net Present Value -1481737.45 By analyzing over the given table, it can be observed that the net present value comes out to be positive. Hence in such case it is suggested for Shine Plc to accept the new security light project because it is giving positive net present value (Khan Jain, 2007). IRR (Internal Rate of Return) of New Product In the above stated formula, net present value at lower rate is the amount which is calculated by subtraction of multiplication of lower present value factor and the cash flows of the project, and initial investment. While present value at lower rate is the amount which can be calculated by multiplying the cash flows of the project by lower present value factor. Here present value at higher rate can be calculated by multiplying cash flows of the project with higher present value factor (Peterson Fabozzi, 2002). In the given case internal rate of return can be calculated as: year earnings after tax depreciation earnings after tax before depreciation PV factor @ 13% Present Value PV factor @ 15% present value initial outlay -1500000 -1500000 -1500000 -1500000 1 -610000 250000 -360000 0.885 -318600 0.876 -315360 2 61250 250000 311250 0.7831 243739.875 0.7561 235336.125 3 19250 250000 269250 0.6931 186617.175 0.6575 177031.875 4 -462000 250000 -212000 0.6133 -130019.6 0.5718 -121221.6 cash flow -1508500 total present value -18262.55 -24213.6 Net Present Value -1481737.45 1475786.4 present value@13%- present Value @ 15% -2957523.85 internal rate of return 14% Internal rate of return helps an organization in selection of the best project from the various alternatives. If the organization has internal rate of return less than the required rate of return then the companys cost of capital, then the project should be accepted. While if the company has greater internal rate of return than companys cost of capital, the project should be rejected. In calculation of IRR, shine plc has assumed 15% rate of return (Cristodoulou). Hence by analyzing the company internal rate of return it can be said that, shine plc has a negative internal rate if return of 14%. The major limitation with internal rate of return is that it does not consider the future cash flows. Hence by considering internal rate of return, it is recommended for shine Plc to not to accept the project. Report to Senior Management Regarding Project By analyzing the above data and calculations, it can be observed that the company has negative cash flows. However the company has positive net present value. Hence by analyzing the net present value it can be said that the project should be accepted. By comparing between shine plc internal rate of return which is, 14% and its cost of capital which is given in the case study as 13%. It can be said that the project should not to be accepted. It is due to negative internal rate of return. Hence it can be said that by investing in new security light, the company will not get profitable results (Bierman Smidth2012). Forward Exchange Requirement Forward contract is an agreement between two parties, in which there is an obligation on the part of buyer to purchase the asset and on the part of seller to sell the product at a fixed determined price. This type of contracting is usually done for long term basis to save both the firms from uncertainty. Besides this, the use of forward contract is higher in case of foreign contracts. This is done due to difference in the structure of the economy, politics, market and many more. It is usually done to save both the parties from foreign currency exchange rates and any market fluctuations (Delaney Whittington, 2007). The US firm has decided to buy the products of shine plc at $3.4 million. The payment would be made to the shine plc company in three months time. However it has been estimated by shine Plc that the cost of this forward contract to the company would be 3.05 million. By observing the whole scenario regarding forward contract, it can be said that the company would arrive at a profit of 0.15. The profit is arrived by subtracting $3.4million as converted into 3.20 million and 3.05. Hence by analyzing over the forward contracting transaction, it can be said that the company has done profitable transaction, and company should accept the offer of US firm. Besides this the company has observed that the bank will pay Euros in return of dollar to company as 3.25, while the company has demanded 3.18. Hence by this it can be analyzed that the company is at liquidity position, as it helps the company in earning profit of 0.07 Euros. Conclusion By analyzing the given assignment, it can be concluded that the new security light should be considered by the company, and it should not accepted. This is due to negative cash flows of the company. Besides this the company has negative net present value and negative internal rate of return. This shows that the company will not get any profitable return as expected by investing in the new project of new security light. While analyzing the forward contract as done between the U.S. firm and the shine plc, it can be concluded that the company has made a good decision. It is due to having a profitable amount of 0.15 million. Hence at last it can be said that the company should reject the proposal of new security line and accept the proposal of U.S. firm on the basis of forward contract. References Bierman, H Smidth, J, S,. (2012) The capital budgeting decision: Economic analysis of investment projects, edition 9th, Routledge, Abingdon Brigham, E Ehrhardt, M,. (2007) Financial management: Theory Practice, Thomson, USA Cristodoulou, A,. The internal rate of return problems and matters of solution. Retrieved on https://www.iamb.it/share/img_new_medit_articoli/802_32cristodoulou.pdf Delaney, P, R Whittington, O,R,. (2007) Wiley CPA examination review 2007-2008, problems and solutions, John Wiley sons, Canada Khan Jain,. (2007) Financial management, Tata McGraw hill, New Delhi Lasher, W, R,. (2016) Practical financial management, Cengage learning, USA Moyer, R, C,. McGuigan, J, R Rao, R, O.P. (2014) Contemporary financial management, Cengage learning, USA Peterson, P,P Fabozzi, F, J,. (2002) Capital budgeting: Theory and practice, John Wiley sons, Canada Shapiro,. (2008) Capital budgeting and Investment analysis, Pearson education India

Mathematics Portfolio Sl Essay Example

Mathematics Portfolio Sl Essay Mathematics Standard Level Teacher: Mr. Lazaro Name: Fatema Ismailjee IB 1 2011 Sequence is a set of things (usually numbers) that are in order. e. g. 1, 2, 3, 4, Where 1 is the first term, 2 is the second term and so on. ( in the end means that the sequence goes on forever. Three dots in the middle e. g. 1, 2, 3 7, 8 indicate that the pattern continues until the next number appears. There is finite and infinite sequence, infinite sequence is when the sequence has no end and finite is a set with a function e. g. {1, 3, n} Calculating specific terms leads to an nth  term formula. Before creating a rule of calculation, you need to realize that sequences are functions with the specific domain of the counting numbers {1, 2, 3, 4, 5, }. So the n replaces x  as the input variable and instead of writing  y, we use  an  as the output variable.Arithmetic sequence: the difference between one term and the next is a constant in arithmetic sequence. The general formula is an  = a1à ‚  + (n 1) d Geometric sequence: A geometric sequence is a group of numbers where each term after the first is found by multiplying the previous one by a fixed non zero number called common ratio. The general formula is an = a1 ? rn-1 Series is the sum of terms of a sequence. Sn = x1 + x2 +. xn Arithmetic series: The general formula is Sn  = n/2(a1  + an) Geometric series: a series which has a constant ratio between terms.The general formula is Sn = a1 (1 – rn) 1 r TRIANGULAR NUMBERS Triangular number is the number of dots in an equilateral triangle uniformly filled with dots. This is an investigation task whereby I will try to find number of shapes of geometric figures which form triangular numbers. I will use different sources of information to attain shapes and figures. For the calculations required, different math techniques will be used for the different shape obtained. Aim In this task I will consider geometric shapes which lead to special numbers.The simplest exa mples of these are square numbers, 1, 4, 9, 16, which can be represented by squares of side 1, 2, 3 and 4. The following diagrams show a triangular pattern of evenly spaced dots. The numbers of dots in each diagram are examples of triangular numbers (1, 3, 6, ). .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 6 10 15 There is a sequence of the number of dots in the triangular shape above.Complete the triangular sequence with three more terms. . . . . . . . . . . . . . . . . . . . . . 21 dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 dots Find a general statement that represents the nth triangular number in terms of n. In words: The top row has one dot and each successive row under it has one more dot.Using the formula: 1. Find the common difference between the numbers in the sequence. 2. Use the general formula tn = an2 + bn + c. 3. Three equations will be forme d. Using the elimination method find the coefficients i. e. a, b and c. 4. Substitute in the general formula. The general statement can be reached by following the steps above. Common difference: d= U2 – U1 = U3 – U2 1, 3, 6, 10, 15, d= 3-1 = 2 6-3 = 3 10-6 = 4 15-10 = 5 d= 3-2 = 1 4-3 = 1 5-4 = 1 The difference in terms is found in the second stage so the formula will be n2 . 2 Testing: n = 1 , triangular number = 1 22 = 12 1 12 = 12 n = 2, triangular number = 3 n22 = 92 3 92 = 32 32 12 = 12 12 so this will be 12 n therefore, 2 12 n2 = 12n As the common difference in the second stage is 1, it can be deduced that the formula for the nth term is a quadratic equation. I will use the general formula to find the nth term, tn = an2 + bn + c where a and b are the coefficients and c is constant and n is the number of term. 2 n2 = 12 n = n2 + n 2 When n = 1 1 = a (1)2 + b (1) + c 1 = a + b + c . (i) n = 2 3 = a (2)2 + b (2) + c 3 = 4a + 2b + c . (ii) n = 3 6 = a (3)2 + b (3 ) + c 6 = 9a + 3b + c . (iii) Using the elimination method: 3 = 4a + 2b + c 6 = 9a + 3b + c 1 = a + b + c 3 = 4a + 2b + c 2 = 3a + b 3 = 5a + b Now that two equations are obtained: To find the variables i. e. a, b one of them is eliminated.In this case the equations are being subtracted. b will be eliminated first in order to find a. Substitute the values of a in the equation to find the value of b. 3 = 5a + b 3 = 5a + b 1 = a + b + c 2 = 3a + b 3 = 5(1 ) + b 1 = 1 + 1 +c 2 2 2 1 = 2a 3 5 = b 2 2 2 a = 1 b = 1 c = 0 2 Therefore the formula for finding the nth term will be as follows: tn = 1n2 + 1n 2 2 tn = n2 + n 2 Use of technology to find the general statement: Calculator used: CASIO fx-9750 GA PLUS n| 1| 2| 3| 4| 5| 6| 7| y| 1| 3| 6| 10| 15| 21| 28| Let n = x 1. Select STAT. 2. Encode values for x in list 1 and for y in list 2. 3. Select GRPH (by pressing F1). 4. Select GPH1 (by pressing F1 again). 5. Select x^2 (by pressing F3). The display will show: a = 1 2 b = 1 c = 0 y = ax 2 + bx + c 1n2 + 1n y = 2 2 = n2 + n 2 Consider stellar (star) shapes with p vertices, leading to p-stellar numbers. The first four representations for a star with six vertices are shown in the four stages S1 – S4 below. The 6-stellar number at each stage is the total number of dots in the diagram. Find the number of dots (i. e. the stellar number) in each stage up to S6. Stellar numbers are figurate number, based on the number of dots of units that can fit in a centred hexagon or star shapes. S1 – S4 are the numbers of dots in the stars.To find up to S6 find the common difference (d) followed by the addition of numbers of star in the previous star. S1 has 1 dot S2 has 13 dots S3 has 37 dots S4 has 73 dots Find the common difference between the terms. d = S2 – S1 S3 – S2 d = 13 – 1 = 12 37 – 13 = 24 73 – 37 = 36 As the difference is not constant, find the difference within the answers. d = 36 – 24 = 12 13 – 12 = 12 The c ommon difference is 12. S 5 = 36 + 12 = 48 73 + 48 = 121dots S6 = 48 + 12 = 60 121 + 60 = 181dots Find an expression for the 6-stellar number at stage S7. As shown above, the common difference is 12.As it’s a sequence it follows the same trend therefore: To find the next number of dots in the sequence, add it with 12 first and from the second star add it with the multiples of 12, i. e. 24, 36, 48 etc. S6 = 48 + 12 = 60 = S5 + 60 = 121 + 60 = 181 S7 = 60 + 12 = 72 = S6 + 72 = 181 + 72 = 253 S7 = 253 Find a general statement for the 6-stellar number at stage Sn in terms of n. Use the same general formula to obtain the three equations: The general formula: tn = an2 + bn + c When n = 1 1 = a (1)2 + b (1) + c 1 = a + b + c . (i) n = 2 13 = a (2)2 + b (2) + c 13 = 4a + 2b + c . (ii) = 3 37 = a (3)2 + b (3) + c 37 = 9a + 3b + c . (iii) Using the elimination method: 37 = 9a + 3b + c 13 = 4a + 2b + c 13 = 4a + 2b + c 1 = a + b + c 24 = 5a + b 12 = 3a + b After attaining two equations, either of the coefficients should be eliminated. b in this case which will lead us to find a. Substitute value of a in the equation to find b. Hence, substitute values of a and b for c. 24 = 5a + b 24 = 5a + b 1 = a + b + c 2 = 3a + b 24 = 5(6) + b 1 = 6 + (-6) + c 12= 2a 24 – 30 = b 1 – 0 = c 2 2 a = 6 b = -6 c = 1 Substitute a, b and c in the general statement. General statement: tn = 6n2 – 6n + 1 Now repeat the steps above for other values of p Considering stellar (star) shapes when p=7 and when p=8 leading to p-stellar numbers. p = 7 Find the number of dots (i. e. the stellar number) in each stage up to S6. S1 has 1 dot S2 has 15 dots S3 has 43 dots S4 has 85 dots d = 15 – 1 = 14 3 – 15 = 28 85 – 43 = 42 d = 42 – 28 = 14 28 – 14 = 14 e. g. S4 = 43 + 14 = 42 S3 + 42 43 + 42 = 85 dots S 5 = 42 +14 =56 S4 + 56 85 + 56 = 141dots S6 = 56 +14 = 70 S5 + 70 141 + 70 = 211dots Find an expression for the 6-stellar number at stage S7 . As shown above, the common difference is 14. As it’s a sequence it follows the same trend therefore: To find the next number of dots in the sequence, add it with 2 first and from the second star add it with the multiples of 2, i. e. 14, 28, 42 etc. S7 = 70 + 14 = 84 S6 + 84 211 + 84 = 2955dots S7 = 295dots Find a general statement for the 6-stellar number at stage Sn in terms of n.To find the three equations, use the general formula tn = an2 + bn + c. When n = 1 1 = a (1)2 + b (1) + c 1 = a + b + c . (i) n = 2 15 = a (2)2 + b (2) + c 15 = 4a + 2b + c . (ii) n = 3 43 = a (3)2 + b (3) + c 43 = 9a + 3b + c . (iii) Three equations are obtained, to find a, b and c, the equations need to be solved. Elimination method is one of the ways from which we can attain the coefficients and constant. Using elimination method: Firstly, we need to remain with two equations at the end so subtract equations (equation iii – ii and equation ii – i) and two will be remained. 3 = 9a + 3b + c 15 = 4a + 2b + c 15 = 4a + 2b + c 1 = a + b + c 28 = 5a + b 14 = 3a + b Now that there are two equations, find a and b. Subtract the equation to eliminate one variable. After one is found, the other can be easily found by substituting the value of variable attained in the equation. 28 = 5a + b 28 = 5a + b 1 = a + b + c 14 = 3a + b 28 = 5(7) + b 1 = 7 + (-7) + c 4= 2a 28 – 35 = b 1 – 0 = c 2 2 a = 7 b = -7 c = 1 Substitute a, b and c in the general statement. General statement: tn = 7n2 – 7n + 1 p = 8 S1 has 1 dot S2 has 17 dots S3 has 49 dots S4 has 97 dots Find the common difference: d = 17 – 1 = 16 49 – 17 = 32 97 – 49 = 48 As the difference is not constant, subtract the answers to find the common difference. d = 32 – 16 = 16 48 – 32 = 16 To find the following number in the star e. g. S4 = 32 + 16 = 48 S3 + 48 49 + 48 = 97 dots The common difference is 16.If observed carefully the number is found by adding it with mu ltiples of 16 i. e. 32, 48, 64, 80 etc. S 5 = 48 +16 = 64 S4 + 64 97 + 64 = 161dots S6 = 64 + 16 = 80 S5 + 80 161 + 80 = 241dots Find an expression for the 6-stellar number at stage S7. As shown above, the common difference is 16. As it’s a sequence it follows the same trend therefore: To find the next number of dots in the sequence, add it with 16 first and from the second star add it with the multiples of 3, i. e. 32, 48, 64 etc. S7 = 80 + 16 = 96 S6 + 96 241 + 96 = 337dots S7 = 337dots Find a general statement for the 6-stellar number at stage Sn in terms of n.I will use the same general formula to obtain the three equations: The general formula: tn = an2 + bn + c When n = 1 1 = a (1)2 + b (1) + c 1 = a + b + c . (i) n = 2 17 = a (2)2 + b (2) + c 17 = 4a + 2b + c . (ii) n = 3 49 = a (3)2 + b (3) + c 49 = 9a + 3b + c . (iii) Using the elimination method: 49 = 9a + 3b + c 17 = 4a + 2b + c 17 = 4a + 2b + c 1 = a + b + c 32 = 5a + b 16 = 3a + bNow there are two equations, so b has to eliminated by subtracting the two equations to find a. Once, a is obtained one of the equation has to be chosen and substitute the value of a in it. Hence b is obtained. 32 = 5a + b 32 = 5a + b 1 = a + b + c 16 = 3a + b 32 = 5(8) + b 1 = 3 + (-3) + c 16 = 2a 32 – 40 = b 2 2 a = 8 b = -8 c = 1 Substitute a, b and c in the general statement. General statement: tn = 8n2 – 8n + 1Hence, produce the general statement, in terms of p and n,that generates the sequence of p-stellar numbers for any value of p at stage Sn. The general statements produced are: tn = 6n2 – 6n + 1 tn = 7n2 – 7n + 1 tn = 8n2 – 8n + 1 I observed it, and reached to the conclusion that for all the three statements the number of p and number of coefficient that is a and b is the same. Therefore the general statement in terms of p and n that generates the sequence of p-stellar numbers for any value of p at stage Sn is: tn = pn2 – pn + 1 Test the validity of the general st atement. When p = 5 S1 has 1 dot S2 has 11 dots S3 has 31 dotsS4 has 61 dots Common difference: d = 11 – 1 = 10 31 – 11 = 20 61 – 31 = 30 As the difference is not constant, subtract it within the answer obtained: d = 30 – 20 = 10 20 – 10 = 10 As seen, the common difference is 5. As it’s a sequence it follows the same trend therefore: To find the next number in the sequence, add it with multiples of 5 i. e. 10, 15, 20, etc. S 5 = 30 + 10 = 40 61 + 40 = 101dots S6 = 40 + 10 = 50 = S5 + 50 = 101 + 50 = 151dots S7 = 50 + 10 = 60 = S6 + 60 = 151 + 60 = 211 S7 = 211dots The general formula: tn = an2 + bn + c When n = 1 1 = a (1)2 + b (1) + c 1 = a + b + c . (i) n = 2 1 = a (2)2 + b (2) + c 11 = 4a + 2b + c . (ii) n = 3 31 = a (3)2 + b (3) + c 31 = 9a + 3b + c . (iii) Using the elimination method: 31 = 9a + 3b + c 11 = 4a + 2b + c 11 = 4a + 2b + c 1 = a + b + c 20 = 5a + b 10 = 3a + b 20 = 5a + b 20 = 5a + b 1 = a + b + c 10 = 3a + b 20 = 5(5) + b 1 = 6 + (-6) + c 10= 2a 20 – 25 = b 1 – 0 = c 2 a = 5 b = -5 c = 1 Substitute a, b and c in the general statement. General statement: tn = 5n2 – 5n + 1 Below are the values of the formula Sn= 5n2 5n +1 n = Stage number in the 5-stellar shape. | y= total number of dots at stage ‘n’. | 1| 1| 2| 11| 3| 31| 4| 61| 5| 101| 6| 151| 7| 211| If we want to know the number of dots in the 5th term using the formula we replace n with 5 based on the formula Sn= 5n2 – 5n + 1 Sn= 5n2 – 5n + 1 Sn= 5 (5)2 – 5 (5) +1 Sn= 101 Limitation: * The value of p should be greater than or equal to 4 i. e. p ? 4. The value of p cannot be negative. It must be a positive integer. The general statement has some limitations as listed above. It is an arithmetic series as seen. It is derived from the equations generated in the 5, 6, 7-stellar shape. The coefficients in each question are equal to the corresponding stellar number p. References: Sequences.   Math Is Fun Maths Resources. Web. 12 Mar. 2011. Help for a Generic Formula for a Stellar Pattern.? Yahoo! Answers.   Yahoo! Answers Home. Web. 12 Mar. 2011. Triangular Number ENotes. com Reference.   ENotes Literature Study Guides, Lesson Plans, and More. Web. 11 Mar. 2011.